The solution of the differential equation (x^2 + xy)dy = (x^2 + y^2)dx is Option 1: y + _e, (y-2)/x, = c, (c is an arbitrary constant) Option 2: y/x + _e, ((y-x)^2)/x, = c, (c is an arbitrary constant) Option 3: y - _e, (y-x)/x, = c, (c is an arbitrary constant) Option 4: y^2 + _e, ((y-x)^2)/x, = c, (c is an arbitrary constant)

Answer: y/x + _e, ((y-x)^2)/x, = c, (c is an arbitrary constant). Solution: Solution Coming Soon

Calculus

Differential Equations

Medium

core

The solution of the differential equation $(x^2 + xy)dy = (x^2 + y^2)dx$ is

$y + \log_e\left|\frac{y-2}{x}\right| = c$ , ( $c$ is an arbitrary constant)

$\frac{y}{x} + \log_e\left|\frac{(y-x)^2}{x}\right| = c$ , ( $c$ is an arbitrary constant)

$y - \log_e\left|\frac{y-x}{x}\right| = c$ , ( $c$ is an arbitrary constant)

$y^2 + \log_e\left|\frac{(y-x)^2}{x}\right| = c$ , ( $c$ is an arbitrary constant)

✅ Correct Option: 2

Solution Coming Soon

2026: 21 May Shift 2

Match the differential equation (List-I) with its order and degree (List-II)

LIST-I		LIST-II
A. $\dfrac{d^3y}{dx^3} + x^2\left(\dfrac{d^2y}{dx^2}\right) = 0$		I. order =3; degree is not defined
B. $\dfrac{d^3y}{dx^3} + y^2 + e^{\frac{dy}{dx}} = 0$		II. order =3; degree=1
C. $1 - \left(\dfrac{dy}{dx}\right)^2 = \left(a\dfrac{d^2y}{dx^2}\right)^{1/3}$		III. order =2; degree=1
D. $y = x\left(\dfrac{dy}{dx}\right)^3 + \left(\dfrac{d^2y}{dx^2}\right)^4$		IV. order =2; degree=4

Choose the correct answer from the options given below:

2026: 19 May Shift 1

Match List-I with List-II

List-I (Differential equation)	List-II (The sum of degree and order of the differential equation)
(A) $x\frac{d^2y}{dx^2} = \left(1+\left(\frac{dy}{dx}\right)^2\right)^{\frac{1}{3}}$	(I) 2
(B) $x\frac{d^2y}{dx^2} = \left(2-\frac{dy}{dx}\right)^{\frac{1}{7}}$	(II) 9
(C) $3+\frac{dy}{dx} = y+\frac{d^3y}{dx^3}$	(III) 4
(D) $\frac{dy}{dx} = x+3$	(IV) 5

Choose the correct answer from the options given below:

2026: 19 May Shift 1

The general solution of the differential equation $\frac{dy}{dx} + y\cot x = 2\cos x$ is (Where C is an arbitrary constant)

2025: 2 June Shift 1

The solution of the differential equation $ydx + (x - y^2)dy = 0$ is

2023: 30 May Shift 3

If m and n are respectively the order and degree of the differential equation :

$\left(\frac{d^2 y}{dx^2}\right)^5 + 6 \frac{\left(\frac{d^2 y}{dx^2}\right)^3}{\frac{d^3 y}{dx^3}} + \frac{d^3 y}{dx^3} = x^2 + 5$ , then :

2022: 6 Aug Shift 2

Match List I with List II

List I	List II
A. The number of arbitrary constants in the particular solution of differential equation of order 2	I. 1
B. The number of arbitrary constants in the general solution of differential equation of order 2	II. 0
C. The integrating factor of differential equation $\frac{dy}{dx} + \frac{1}{x}y = 3, x > 0$ , is	III. 2
D. For differential equation, $x^2 \frac{dy}{dx} + x = xy, x > 0, \lim_{x \to 0^+} y(x)$ is equal to	IV. $x$

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